Báo cáo toán hoc:"Tiling tripartite graphs with 3-colorable graphs" pps

Tiling tripartite graphs with 3-colorable graphs
Ryan Martin
∗
Iowa State University
Ames, IA 50010
Yi Zhao
†
Georgia State University
Atlanta, GA 30303
Submitted: Apr 25, 2008; Accepted: Aug 22, 2009; Published: Aug 31, 2009
Abstract
For any positive real number γ and any positive integer h, there is N
0
such that
the following holds. Let N  N
0
be such that N is divisible by h. If G is a tripartite
graph with N vertices in each vertex class such that every vertex is adjacent to at
least (2/3 + γ)N vertices in each of the other classes, then G can be tiled perfectly
by copies of K
h,h,h
. This extends the work in [Discrete Math. 254 (2002), 289-
308] and also gives a su fficient condition for tiling by any fixed 3-colorable graph.
Furthermore, we show that the minimum-degree (2/3 + γ)N in our result cannot
be replaced by 2N/3 + h − 2.
1 Introdu ction
Let H be a graph on h vertices, and let G be a graph on n vertices. Tiling (or pack-
ing) problems in extremal graph theory are investigations of conditions under which G
must contain many vertex disjoint copies of H (as subgraphs), where minimum degree
conditions are studied the most. An H-tiling of G is a subgraph of G which consists of
vertex-disjoint copies of H. A pe rfect H-tiling, or H-factor, of G is an H-tiling consisting

of ⌊n/h⌋ copies of H. A very early tiling result is implied by Dirac’s theorem on Hamilton
cycles [6], which implies that every n- vertex graph G with minimum degree δ(G)  n/2
contains a perfect matching (usually called 1-factor, instead of K
2
-factor). Later Corr´adi
and Hajnal [4] studied the minimum degree of G that guarantees a K
3
-factor. Hajnal
and Szemer´edi [9] settled the tiling problem for any complete graph K
h
by showing that
∗
Corresponding author. Research supported in part by NSA grants H98230-05-1-02 57 and H98230 -
08-1- 0015. Email:
†
Research supported in part by NSA grants H98230-05-1-0079 and H98230-07-1-0019. Part of this
research was done while working at University of Illinois at Chicago. Email:
the electronic journal of combinatorics 16 (2009), #R109 1
every n-vertex graph G with δ(G)  (h − 1)n/h contains a K
r
-factor (it is easy to see
that this is sharp). Using the celebrated Regularity Lemma of Szemer´edi [23], Alon and
Yuster [1, 2] generalized the above tiling results for arbitrary H. Their theorems were
later sharpened by various researchers [14, 12, 22, 17]. Results and methods for tiling
problems can be found in a recent survey of K¨uhn and Osthus [18].
In this paper, we consider multipartite tiling, which restricts G to be an r-partite graph.
When r = 2, The K¨onig-Hall Theorem (e.g. see [3]) provides necessary and sufficient
conditions to solve the 1-factor problem for bipartite graphs. Wang [24] considered K
s,s
-

factors in bipartite graphs for all s > 1, the second author [25] gave the best possible
minimum degree condition for this problem. Recently Hladk´y and Schacht [10] determined
the minimum degree threshold for K
s,t
-factors with s < t.
Let G
r
(N) denote the family of r-partite graphs with N vertices in each of its partition
sets. In an r-part ite graph G, we use
¯
δ(G) for the minimum degree from a vertex in one
partition set to any other partition set. Fischer [8] proved almost perfect K
3
-tilings in
G
3
(N) with
¯
δ(G)  2N/3 and Johansson [11] gives a K
3
-factor with the less stringent
degree condition
¯
δ(G)  2N/3 + O(
√
N).
For general r > 2, Fischer [8] conjectured the following r-partite version of the Hajnal–
Szemer´edi Theorem: if G ∈ G
r
(N) satisfies

¯
δ(G)  (r − 1)N/r, then G contains a K
r
-
factor. The first author and Szemer´edi [20] proved this conjecture for r = 4. Csaba and
Mydlarz [5] recently proved that the conclusion in Fischer’s conjecture holds if
¯
δ(G) 
k
r
k
r
+1
n, where k
r
= r + O(log r). On the other hand, Magyar and the first author [19]
showed that F ischer’s conjecture is false for all odd r  3: they constructed r-partite
graphs Γ(N) ∈ G
r
(N) for infinitely many N such that
¯
δ(Γ(N) ) = (r − 1)N/r and yet
Γ(N) contains no K
r
-factor. Nevertheless, Magyar and the first author proved a theorem
(Theorem 1.2 in [19]) which implies the following Corr´adi-Hajnal-type theorem.
Theorem 1.1 ([19]) If G ∈ G
3
(N) satisfies
¯

δ(G)  (2/3)N + 1, then G contains a
K
3
-factor.
In this paper we extend this result to all 3-colorable graphs. Our main result is on K
h,h,h
-
tiling.
Theorem 1.2 For any positive real number γ and any positive integer h, there is N
0
such
that the following holds. Given an integer N  N
0
such that N is divisible by h, if G is a
tripartite g raph with N vertices in each vertex clas s such that every vertex is adjacent to
at least (2/3 + γ)N vertices in each of the other clas s es, then G contains a K
h,h,h
-factor.
Since the complete tripartite graph K
h,h,h
can be perfectly tiled by any 3-colorable graph
on h vertices, we have the following corollary.
the electronic journal of combinatorics 16 (2009), #R109 2
Corollary 1.3 Let H be a 3-colorable graph of order h. For any γ > 0 there exi s ts a
positive integer N
0
such that if N  N
0
and N is divisibl e by h, then every G ∈ G
3

(N)
with
¯
δ(G)  (2/3 + γ)N contains an H-factor.
The Alon–Yuster theorem [2] says that for any γ > 0 and any r-colorable graph H there
exists n
0
such that every graph G of order n  n
0
contains an H-factor if n divisible by h
and δ(G)  (1 − 1/r)n + γn (Koml´os, S´ark¨ozy and Szemer´edi [14] later reduced γn to a
constant that depends only on H). Corollary 1.3 gives another proof of this theorem for
r = 3 as follows. Let G be a graph of order n = 3 N with δ(G)  2n/3 + 2γn. A random
balanced partition of V (G) yields a subgraph G
′
∈ G
3
(N) with
¯
δ(G
′
)  δ(G)/3 −o(n) 
(2/3 + γ)N. We then apply Corollary 1.3 to G
′
obtaining an H-f actor in G
′
, hence in G.
Instead of proving Theorem 1.2, we actually prove the stronger Theorem 1.4 below. Given
γ > 0, we say that G =


V
(1)
, V
(2)
, V
(3)
; E

∈ G
3
(N) is in the extreme case with parameter
γ if there are three sets A
1
, A
2
, A
3
such that A
i
⊆ V
(i)
, |A
i
| = ⌊N/3⌋ for all i and
d(A
i
, A
j
) :=
e(A

i
, A
j
)
|A
i
||A
j
|
 γ
for i = j. If G ∈ G
3
(N) satisfies
¯
δ(G)  (2/3 + γ)N, then G is not in the extreme case
with para meter γ. In fact, any two sets A and B of size ⌊N/3⌋ from two different vertex
classes satisfy deg(a, B)  γN, for all a ∈ A, and consequently d(A, B) > γ. Theorem 1.2
thus follows from Theorem 1.4, which is even stronger b ecause o f its weaker assumption
¯
δ(G)  (2/3 −ε)N.
Theorem 1.4 Given any positive integer h and any γ > 0, there exis ts an ε > 0 and
an integer N
0
such that whenever N  N
0
, and h divides N, the following holds: If
G ∈ G
3
(N) satisfies
¯

δ(G)  (2/3 −ε)N, then either G contains a K
h,h,h
-factor or G is in
the extreme case with parameter γ.
The following proposition shows that the minimum degree
¯
δ(G)  (2/3 + γ)N in Theo-
rem 1 .2 cannot be replaced by 2N/3 + h −2.
Proposition 1.5 Given any positive integer h  2, there exists an integer q
0
such that
for any q  q
0
, there exists a tripartite graph G
0
∈ G
3
(N) with N = 3qh such that
¯
δ(G
0
) = 2qh + (h −2) and G
0
has no K
h,h,h
-factor.
The structure of the paper is as follows. We first prove Propo sition 1.5 in Section 2. After
stating the Regularity Lemma and Blow-up Lemma in Section 3, we prove Theorem 1.4
in Section 4. We give concluding remarks and open problems in Section 5.
the electronic journal of combinatorics 16 (2009), #R109 3

2 Proo f of Proposition 1.5
In a tr ipart ite graph G = (A, B, C; E), the gr aphs induced by (A, B), (A, C) and (B, C)
are called the natural bipartite subgraphs of G. First we need to construct a balanced
tripartite K
3
-free graph in which all na tura l bipartite graphs are regular and C
4
-free. Our
construction below is based a construction in [25] of sparse regular bipartite graphs with
no C
4
.
Lemma 2.1 For each in teger d  0, there exists an n
0
such that, if n  n
0
, there exists
a balanced tripartite graph, Q(n, d) on 3n vertices such that each of the 3 natural bipartite
subgraphs are d-regular, C
4
-free and triangle-free.
Proof. A Sidon set is a set of integers such that sums i + j are distinct for distinct pairs
i, j from the set. Let [n] = {1, . . . , n}. It is well known (e.g., [7]) that [n] contains a Sidon
set of size about
√
n for large n. Suppose that n is sufficiently large. Let S be a d-element
Sidon subset of [
n
3
−1]. Given two copies of [n], A and B, we construct a bipartite graph

P (A, B) on (A, B) whose edges are (ordered) pairs ab, a ∈ A, b ∈ B such that b −a (mod
n) ∈ S. It is shown in [25] (in the proo f of Propo sition 1.3) that P(A, B) is d-regular with
no C
4
. Given three copies of [n], A, B and C, let Q be the union of P(A, B), P (B, C)
and P(C, A). In order to show that Q is the desired graph Q(n, d), we need to verify that
Q is K
3
-free. In fact, if a ∈ A, b ∈ B, and c ∈ C form a K
3
, then there exist i, j, k ∈ S
such that
b ≡ a + i, c ≡ b + j, a ≡ c + k (mod n),
which implies that i + j + k ≡ 0 mod n. But this is impossible for i, j, k ∈ [
n
3
− 1]. 
Proof of Proposition 1.5. We will construct 9 disjoint sets A
(i)
j
with i, j ∈ {1, 2, 3} .
The union A
(i)
1
∪ A
(i)
2
∪ A
(i)
3

defines the i
th
vertex-class, while the triple (A
(1)
j
, A
(2)
j
, A
(3)
j
)
defines the j
th
column.
Construct G
0
as follows: For i = 1, 2, 3, let |A
(i)
1
| = qh −1 , |A
(i)
2
| = qh and |A
(i)
3
| = qh + 1.
Let the graph in column 1 be Q(qh − 1, h − 3) (as given by Lemma 2.1), the graph in
column 2 be Q(qh, h −2) and the graph in column 3 be Q(qh + 1, h − 1). If two vertices
are in different columns and different vertex-classes, t hen they are also adjacent. It is

easy to verify that
¯
δ(G
0
) = 2qh + (h −2).
Suppose, by way of contradiction, that G
0
has a K
h,h,h
-factor. Since there are no triangles
and no C
4
’s in any column, the intersection of a copy of K
h,h,h
with a column is either a
star, with all leaves in the same vertex-class, or a set of vertices in the same vertex-class.
So, each copy of K
h,h,h
has at most h vertices in column 3. A K
h,h,h
-factor has exactly
3q copies of K
h,h,h
and so the factor has at most 3qh vertices in column 3. But there are
3qh + 3 vertices in column 3, a contradiction. 
the electronic journal of combinatorics 16 (2009), #R109 4
3 The Regularity Lemma and Blow-up Lemma
The Regularity Lemma and the Blow-up Lemma are main tools in the proof of Theo-
rem 1 .4. Let us first define ε-regularity and (ε, δ)-super-regularity.
Definition 3.1 Let ε > 0. Suppose that a graph G contains disjoint vertex-sets A and B.

1. The pair (A, B) is ε-regular if for every X ⊆ A and Y ⊆ B, satisfying |X| >
ε|A|, |Y | > ε|B|, we have |d(X, Y ) − d(A, B)| < ε.
2. The pair (A, B) is (ε, δ)-super-regular if (A, B) is ε-regular and deg(a, B) > δ|B|
for all a ∈ A and deg(b, A) > δ| A| for all b ∈ B.
The celebrated Regularity Lemma of Szemer´edi [23] has a multipartite version (see survey
papers [15, 16]), which guarantees that when applying the lemma to a multipartite graph,
every resulting cluster is from one partition set. Given a vertex v and a vertex set S in a
graph G, we define deg(v, S) as the number of neighbors of v in S.
Lemma 3.2 (Regularity Lemma - Tripartite Version) For every posi tive ε there is
an M = M(ε) such that if G = (V, E) is any tripartite graph with partition sets V
(1)
, V
(2)
,
V
(3)
of si z e N, and d ∈ [0, 1] is any real number, then there are partitions of V
(i)
into
clusters V
(i)
0
, V
(i)
1
, . . . , V
(i)
k
for i = 1, 2, 3 and a subgraph G
′

= (V, E
′
) with the following
properties:
• k  M,
• |V
(i)
0
|  εn for i = 1, 2, 3,
• |V
(i)
j
| = L  εn for all i = 1, 2, 3 and j  1,
• deg
G
′
(v, V
(i
′
)
) > deg
G
(v, V
(i
′
)
) −(d + ε)N for all v ∈ V
(i)
and i = i
′

,
• all pairs (V
(i)
j
, V
(i
′
)
j
′
), i = i
′
, 1  j, j
′
 k, are ε-regular in G
′
, each with density
either 0 or exceeding d.
We will also need the Blow-up Lemma of Koml´os, S´ark¨ozy and Szemer´edi [13].
Lemma 3.3 (Blow-up Lemma) Given a graph R of order r and positive parameters
δ, ∆, there exists an ε > 0 such that the followin g holds: Let N be an arbitrary positive
integer, and let us replace the vertices of R wi th pairwise disjoint N-s ets V
1
, V
2
, . . . , V
r
.
We construct two gra phs on the same vertex-set V =


V
i
. The graph R(N) is obtained
by replacing all edges of R with copies of the complete bi partite graph K
N,N
and a sparser
graph G is constructed by repl acing the edges o f R with som e (ε, δ)-super-regular pairs. If
a graph H with maximum degree ∆(H)  ∆ can be embedded into R(N), then i t can be
embedded into G.
the electronic journal of combinatorics 16 (2009), #R109 5
4 Proo f of Theorem 1.4
In this section we prove Theorem 1.4. First we sketch the proof.
We begin by applying the Regularity Lemma to G, partitio ning each vertex class into ℓ
clusters and an exceptional set. Next we define the cluster graph G
r
(whose vertices a r e
the clusters of G and where clusters from different partition classes are adjacent if the pair
is regular with positive density), which is 3-partite and such that
¯
δ(G
r
) is almost 2ℓ/ 3.
In Step 1, we use the so-called fuzzy tripartite theorem of [19], which states that either
G
r
is in the extreme case (hence G is in the extreme case) or G
r
has a K
3
-factor. Having

assumed that G
r
has a K
3
-factor S = {S
1
, . . . , S
ℓ
}, in Step 2 we move a small amount of
vertices from each cluster to the exceptional sets such that in each S
j
, all three pairs are
sup er-regular and the three clusters have the same size, which is a multiple of h. If we
now were to apply the Blow-up Lemma to each S
j
, then we would o bta in a K
h,h,h
-factor
covering all the non-exceptional vertices of G.
So we need to handle the exceptional sets before applying the Blow-up Lemma. Step 3 is a
step of preprocessing: we set some copies of K
h,h,h
aside such that in Step 5 we can modify
them by replacing 5h vertices from S
1
with 5 h vertices from an S
j
, j  2. The vertices
in these copies of K
h,h,h

are not now in their original clusters. Since these copies of K
h,h,h
are from triangles of G
r
that are not necessarily in S, we may need to move vertices from
other clusters t o the exceptional sets to keep the balance of the three clusters in each S
j
.
For each exceptional vertex v, we will remove a copy of K
h,h,h
which contains v and 2h−1
vertices from some cluster-triangle S
j
(we call this inserting v into S
j
). If this is done
arbitrarily, the remaining vertices of some S
j
may not induce a K
h,h,h
-factor. In Step 4,
we group exceptional vertices into h-element sets such that all h vertices in one h-element
set can be inserted into the same S
j
. As a result, two clusters in some S
j
may have sizes
that differ by a multiple of h. We then remove a few more copies of K
h,h,h
such that the

sizes o f the three clusters of each S
j
are the same and divisible by h. Unfortunately up
to 5h vertices in each exceptional set may not be removed by this approach. In Step 5
we first insert the remaining exceptional vertices into an arbitrary S
j
, j  2, and then
transfer extra vertices from S
j
to S
1
. As a result, three clusters in all S
j
, j  1 have the
same size, which is divisible by h. At the end of Step 5, we apply the Blow-up Lemma to
each S
j
to complete the K
h,h,h
-factor of G. This ends the proof sketch.
Note that our proo f follows the approach in [19], which has a different way of handling
exceptional vertices from the bipartite case [25]. Although a K
h,h,h
-tiling is more complex
than a K
3
-tiling, our proof is not longer than the non-extreme case in [19] because we
take a dvantage o f results f r om [19].
Let us now start the proof. We assume that N is large, and without loss of generality,
assume that γ ≪

1
h
. We find small constants d
1
, ε, and ε
1
such that (actual dependencies
result from Lemmas 4.1 , 4.4, 4.7, and 3.3):
ε
1
≪ 2ε = d
1
≪ γ. (1)
the electronic journal of combinatorics 16 (2009), #R109 6
For simplicity, we will refrain from using floor or ceiling functions when they are not
crucial.
Begin with a tripartite graph G =

V
(1)
, V
(2)
, V
(3)
; E

with


V

(1)


=


V
(2)


=


V
(3)


= N
such that
¯
δ(G)  (2/3 −ε)N. Apply the Regularity Lemma (Lemma 3.2) with ε
1
and d
1
,
partitioning each V
(i)
into ℓ clusters V
(i)
1

, . . . , V
(i)
ℓ
of size L  ε
1
N and an exceptional set
V
(i)
0
of size at most ε
1
N. Later in the proof, the exceptional sets may grow in size, but
will always remain of size O(ε
1
N). We call the vertices in the exceptional sets exceptional
vertices.
Let G
′
be the subgraph of G defined in the Regularity Lemma. We define the reduced
graph (or cluster graph) G
r
to be the 3-partite graph whose vertices are clusters V
(i)
j
j  1, i = 1, 2, 3, and two clusters are adjacent if and only if they form an ε
1
-regular pair
of density at least d
1
in G

′
. We will use the same notation V
(i)
j
for a set in G and a vertex
in G
r
. Let U
(1)
, U
(2)
, U
(3)
denote three partition sets of G
r
. We know that |U
(i)
| = ℓ.
We observe that
¯
δ(G
r
)  (2/3 − 2d
1
)ℓ. In fact, consider a cluster C ∈ U
(i)
and a vertex
x ∈ C, the number m of clusters in U
(i
′

)
(i
′
= i) that are adjacent to C satisfies

2
3
− ε

N − (d
1
+ ε
1
)N  deg
G
(v, V
(i
′
)
) −(d
1
+ ε
1
)N  deg
G
′
(x, V
(i
′
)

)  mL.
Since N  Lℓ and ε + ε
1
 d
1
, we have m  (2/3 −ε −d
1
− ε
1
) ℓ  (2/3 −2d
1
)ℓ.
Assume that G is not in the extreme case with parameter γ. We claim that G
r
is not in
the extreme case with parameter γ/3. Suppose instead, that there are subsets S
i
⊂ U
(i)
,
i = 1, 2, 3, o f size ℓ/3 with density at most γ/3. Let A
i
denote the set of all vertices
of G contained in a cluster of S
i
. Then N(1 − ε
1
)/3  |A
i
| = Lℓ/3  N/3 because

Lℓ  (1 −ε
1
)N. The number of edges of G between A
i
and A
i
′
, i = i
′
, is at most
e
G
(A
i
, A
i
′
)  e
G
′
(A
i
, A
i
′
) + |A
i
|(d
1
+ ε

1
)N 
γ
3

ℓ
3

2
L
2
+ (d
1
+ ε
1
)
N
2
3

2γ
3

N
3

2
,
provided that 9(d
1

+ ε
1
)  γ. After adding at most ε
1
N/3 vertices to each A
i
, we obtain
three subsets of V
(1)
, V
(2)
, V
(3)
of size N/3 with pairwise density at most (2γ/ 3 + ε
1
)  γ
in G.
Step 1: Find a K
3
-factor in G
r
We apply the following result (Theorem 2.1 in [19]) to the reduced graph G
r
with α = γ/3
and β = 2d
1
.
Lemma 4.1 (Fuzzy tripartite theorem [19 ]) For any α > 0, there exist β > 0 and
ℓ
0

, such that the follows holds for all ℓ  ℓ
0
. Every balanced 3-partite graph R ∈ G
3
(ℓ) with
¯
δ(R)  (2/3 − β)ℓ either contain s a K
3
-factor or is in the extreme case with parame ter
α.
the electronic journal of combinatorics 16 (2009), #R109 7
Since G
r
is not in the extreme case with parameter γ/3, it must contain a K
3
-factor
S = {S
1
, S
2
, . . . , S
ℓ
}. After relabeling, we assume that S
j
=

V
(1)
j
, V

(2)
j
, V
(3)
j

for all
j. In G
r
, we call these fixed triangles S
1
, . . . , S
ℓ
columns and consider U
(1)
, U
(2)
, U
(3)
as
rows.
Step 2: Make pairs in S
j
super-regular
For each S
j
, remove a vertex v from a cluster in S
j
and place it in the exceptional set if
v has fewer than (d

1
−ε
1
)L neighbors in one of the other clusters of S
j
. By ε
1
-regularity,
there are a t most 2ε
1
L such vertices in each cluster. Remove more vertices if necessary to
ensure that each non-exceptional cluster is o f the same size and the size is divisible by h.
The Slicing Lemma states the well-known fact that regularity is maintained when small
modifications are made to the clusters:
Proposition 4.2 ( Slicing Lemma, Fact 1.5 in [19]) Let (A, B) be an ε-regular pair
with density d, and, f or some α > ε, let A
′
⊂ A, |A
′
|  α|A|, B
′
⊂ B, |B
′
|  α|B|.
Then (A
′
, B
′
) is an ε
′

-regular pair w i th ε
′
= max{ε/α, 2ε}, and for its density d
′
, we have
|d
′
− d| < ε.
Applying Proposition 4.2 with α = 1 −2ε
1
, any pair of clusters which was ε
1
-regular with
density at least d
1
is now (2ε
1
)-regular with density at least d
1
− ε
1
(because ε
1
< 1/4).
Furthermore, each pair in the cluster-triangles S
j
is (2ε
1
, d
1

−3ε
1
)-super-regular. Each of
the t hree exceptional sets are now of size at most ε
1
N + ℓ ( 2 ε
1
L)  3ε
1
N.
Remark: Because all the pairs in S
j
are super-regular and the complete tripartite graph
on

V
(1)
i
, V
(2)
i
, V
(3)
i

contains a K
h,h,h
-factor, the Blow-up Lemma says that S
j
also con-

tains a K
h,h,h
-factor.
Step 3: Create red copies of K
h,h,h
In this step we show t hat certain triangles exist in G
r
which link each cluster to the one
in S
1
from the same partition class. The purpose of this linking is to be able to handle a
small discrepancy of sizes among the three clusters that comprise S
j
in Step 5.
Definition 4.3 In a tripartite graph R =

U
(1)
, U
(2)
, U
(3)
; E

, one vertex x ∈ U
(1)
(the
cases of x ∈ U
(2)
or U

(3)
are defined accordingly) is r eachable from another vertex
y ∈ U
(1)
in R by using at most 2k triangles, if there is a chain of triangles T
1
, . . . , T
2k
with T
j
=

T
(1)
j
, T
(2)
j
, T
(3)
j

and T
(i)
j
∈ U
(i)
for i = 1, 2, 3 such that the following occurs:
1. x = T
(1)

1
and y = T
(1)
2k
,
the electronic journal of combinatorics 16 (2009), #R109 8
1
1
T
2
T
3
T
4
CC’V
(1)
T
Figure 1: An illustration of how cluster V
(1)
1
is reachable from a cluster C.
2. T
(2)
2j−1
= T
(2)
2j
and T
(3)
2j−1

= T
(3)
2j
, for j = 1, . . . , k, and
3. T
(1)
2j
= T
(1)
2j+1
, for j = 1, . . . , k − 1.
Figure 1 illustrates that V
(1)
1
is reachable from another cluster C by using f our triangles.
The Reachability Lemma (Lemma 2.6 in [19]) says that every cluster of S
1
is reachable
from any other cluster in the same class by using at most four triangles in G
r
. Note
that these triangles are not necessarily the fixed triangles S
j
. The statement of the
Reachability Lemma in [19] refers to the reduced graph, but its proof, in fact, proves the
following general statement:
Lemma 4.4 (Reachability Lemma) For any α > 0, there exist β > 0 and ℓ
0
, such
that the follo wing holds for all ℓ  ℓ

0
. Let R ∈ G
3
(ℓ) be a balanced 3-partite graph with
¯
δ(R)  (2/3 − β)ℓ. Then either each vertex is reachable from every other vertex in the
same class by using at most four triangles or R is in the extreme case with parame ter α.
Let C = V
(1)
1
be a cluster in U
(1)
and let T
1
, T
2
or T
1
, T
2
, T
3
, T
4
be cluster-triangles
which witness that V
(1)
1
is reachable from C by using at most 2k triangles for some
k ∈ {1, 2}. Note that T

1
∩ U
(1)
= S
(1)
1
and either bo t h k = 1 and T
2
∩ U
(1)
= C or k = 2,
T
2
∩ U
(1)
= T
3
∩ U
(1)
= C
′
and T
4
∩ U
(1)
= C.
We need a special case of a well-known embedding lemma in [15], which says that three
reasonably large subsets of three clusters that form a triangle induce a copy of K
h,h,h
.

Proposition 4.5 ( Key Lemma, Theorem 2.1 in [15]) Let ε, d be positive real num-
bers and h, L be positive integers such that (d −ε)
2h
> ε and ε(d −ε)L  h. Suppose that
X
1
, X
2
, X
3
are clusters of size L and any pair of them is ε-regular with density at least
d. Let A
i
⊆ X
i
, i = 1, 2, 3 be three subsets of size at least (d − ε)L. Then (A
1
, A
2
, A
3
)
contains a copy of K
h,h,h
.
If k = 1, then we pick a vertex v ∈ C and apply Proposition 4.5 to find a copy of
K
h,h,h
, called H
′

, in the cluster triangle T
1
such that H
′
∩ V
(2)
and H
′
∩ V
(3)
are in the
the electronic journal of combinatorics 16 (2009), #R109 9
neighb orhood of v. If k = 2, then we first pick a vertex v ∈ C and apply Proposition 4.5
to find a copy of K
h,h,h
, called H
′′
, in the cluster triangle T
3
such that H
′′
∩ V
(2)
and
H
′′
∩ V
(3)
are in the neighborhood of v. Next we pick a vertex v
′

∈ H
′′
∩ V
(1)
(call it
special) and apply Proposition 4.5 to find a copy of K
h,h,h
, called H
′
, in the cluster triangle
T
1
such that H
′
∩ V
(2)
and H
′
∩ V
(3)
are in the neighborhood of v
′
.
Color all of the vertices in H
′
and in H
′′
(if it exists) red and the vertex in C orange. Note
that the special vertex in H
′′

(if existent) is colored red. If a vertex is not colored, we
will heretofore call it uncolored. Repeat t his 5h times fo r each cluster not in S
1
. In this
process all but a constant number of vertices in each cluster remain uncolored since h is
a constant and G
r
consists of a constant number (that is, 3ℓ) of clusters. This is why we
can repeatedly apply Proposition 4.5 ensuring that all the red copies of K
h,h,h
and orange
vertices are vertex-disjoint.
At the end, each cluster not in S
1
has 5h orange vertices (the clusters in S
1
have no orange
vertex). Each cluster has at most 3(ℓ −1)(5h)(h) < 15ℓh
2
red vertices because there are
3(ℓ −1) clusters not in S
1
, the process is iterated 5h times f or each of them and a cluster
gets at most h vertices colored red with each iteration.
Remark: This preprocessing ensures that we may later transfer at most 5h vertices from
any cluster C to S
1
in the following sense: Without loss of generality, suppose C is a
cluster in V
(1)

. In the case when k = 2 (see Figure 1), identify an orange vertex v ∈ C
and its corresponding red subgraphs H
′
and H
′′
, including the special vertex v
′
∈ C
′
. (The
case where k = 1 is similar but simpler.) Recolor v red and uncolor a vertex u ∈ H
′
∩V
(1)
1
.
The red vertices still form two copies of K
h,h,h
, one is H
′
−{u}+{v
′
}, and the other o ne is
H
′′
−{v
′
}+{v}. The number of non-red vertices is decreased by one in C but is increased
by one in V
(1)

1
. We will do this in Step 5.
We now move some uncolored vertices from clusters to the corresponding exceptional
set such that the three clusters in the same column (some S
j
) have the same number
of uncolored vertices. In other words, three clusters in any S
j
are balanced in terms of
uncolored vertices. (Note t hat this number is always divisible by h because the numbers
of red vertices and orange vertices are divisible by h.) Thus, at most 15ℓh
2
vertices
can be removed from a cluster. The three exceptional sets have the same size, at most
3ε
1
N + 15ℓ
2
h
2
 4ε
1
N. Each cluster still has at least (1 − 2ε
1
)L − 15ℓh
2
> (1 − 3ε
1
)L
uncolored vertices.

Step 4: Reduce the sizes of exceptional sets
At present the exceptional sets V
(i)
0
, i = 1, 2, 3, are all of the same size, which is at most
4ε
1
N and divisible by h. Suppose this size is at least 6h. We will remove some copies of
K
h,h,h
from G such that |V
(i)
0
|  5h eventually.
First, we say a vertex v ∈ V
(i)
0
belongs to a cluster V
(i)
j
if deg(v, V
(i
′
)
j
)  d
1
L for all i
′
= i.

Using the minimum-degree condition, for fixed i
′
= i, the number of clusters V
(i
′
)
j
such
the electronic journal of combinatorics 16 (2009), #R109 10
that deg(v, V
(i
′
)
j
) < d
1
L is at most
(1/3 + ε)N
(1 −3ε
1
)L −d
1
L

(1/3 + ε)ℓ
(1 −3ε
1
− d
1
)(1 −ε

1
)
. (2)
Using (1), the expression in (2) is at most (1/3 + d
1
)ℓ. Thus, v is adjacent to at least
d
1
L uncolored vertices in at least (2/3 −d
1
)ℓ clusters in V
(i
′
)
for some i
′
= i. Hence, each
vertex in V
(i)
0
belongs to at least (1/3 −2d
1
)ℓ clusters.
If a vertex v ∈ V
(i)
0
belongs to a cluster V
(i)
j
, then we may insert v into V

(i)
j
(or loo sely
speaking, insert v into S
j
) in the following sense. We permanently remove a copy of K
h,h,h
from G which consists of v, h−1 vertices from V
(i)
j
and h vertices from each of V
(k)
j
, k = i.
Proposition 4.5 guarantees the existence of this K
h,h,h
.
In order to maintain the size of each cluster as a multiple of h, we will bundle exceptional
vertices into h-element sets and handle all h vertices from an h-element set at a time as
follows.
Claim 4.6 Given a subset Y ⊆ V
(i)
0
of at least 3h vertices and a subset U
′
⊆ U
(i)
of at
least (1 − d
1

)ℓ clusters, there are h vertices o f Y that belong to the same cluster C from
U
′
.
Proof. Suppose instead, that at most h −1 vertices of Y belongs to each cluster C ∈ U
′
.
From earlier calculations and the assumption |U
′
|  (1 −d
1
)ℓ, we know that each vertex
of Y belongs to at least (1/3 − 3d
1
)ℓ clusters. By double counting the number of pairs
(v, C) such that v ∈ Y belongs to a cluster C ∈ U
′
, we have
3h

1
3
− 3d
1

ℓ  (h −1)ℓ, (3)
which implies that 9hd
1
 1, contradicting d
1

≪ 1. 
Starting from Y = V
(i)
0
and U
′
= U
(i)
, we apply Claim 4.6 four times to find four disjoint
h-element subsets W
(i)
1
, . . . , W
(i)
4
of V
(i)
0
whose vertices belong to clusters C
(i)
1
, . . . , C
(i)
4
,
respectively. The reason why we need four h-element sets can be seen below when we apply
Lemma 4.7. We can ensure that C
(i)
1
, . . . , C

(i)
4
are different by letting U
′
= U
(i)
\ {C
(i)
j
′
:
j
′
< j} when we select C
(i)
j
.
We now insert W
(i)
j
into C
(i)
j
for i = 1, 2, 3 and j = 1, 2, 3, 4 by removing in total 12h
copies of K
h,h,h
. All of these copies of K
h,h,h
are removed permanently, they will be a part
of the final K

h,h,h
-factor of G. As a result, each C
(i)
j
has h more vertices than the other
two clusters in the same column (unless accidentally more than one C
(i)
j
fall into the same
column).
The Almost-covering Lemma (Lemma 2.2 in [19]) can help us to balance the sizes of each
column:
the electronic journal of combinatorics 16 (2009), #R109 11
Lemma 4.7 (Almost-covering Lemma [19]) For any α > 0, there exist β > 0 and
m
0
, such that the following holds f or all m  m
0
. Let R ∈ G
3
(m) be a balanced 3-partite
graph with
¯
δ(R)  (2/3−β)m. Suppose that T
0
is a partial K
3
-tiling in R with |T | < m −3.
Then, either
1. there exists a partial K

3
-tiling T
′
with |T
′
| > |T | but |T
′
\ T |  15, o r
2. R is in the extreme case with parameter most α.
Let
˜
G be a new 3-partite graph obtained fro m adding four new vertices to each vertex
class of G
r
. The new 12 vertices are clones of the clusters C
(i)
j
for i = 1, 2, 3, j = 1, 2, 3, 4,
and we denote them by
˜
C
(i)
j
. The clones have the same adjacency in G
r
as their originals.
Let m = ℓ + 4 be the size of vertex classes in
˜
G. We have
¯

δ(
˜
G)  (1/3 −3d
1
)m following
from
¯
δ(G
r
)  (1/3 −2d
1
)ℓ.
We apply Lemma 4.7 to
˜
G with α = γ/3, β = 3d
1
, and T = {S
1
, . . . , S
ℓ
} (then |T | <
m −3). The new graph
˜
G is almost the same as G
r
, provided ℓ is large enough, which we
guaranteed when we applied the Regularity Lemma. Thus,
˜
G is not in the extreme case
(otherwise G

r
is in the extreme case). Lemma 4.7 thus provides a larger partial triangle-
cover T
′
with |T
′
\ T |  15. For each triangle T ∈ T
′
\ T , we permanently remove a
copy of K
h,h,h
from the uncolored vertices of T . For each cluster C that is not covered
by the larger T
′
, take an arbitrary set of h uncolored vertices from C and place it into
the exceptional set. As result, all the clusters covered by T
′
∩ T experience no changes
while all other clusters lose h uncolored vertices; therefore the three clusters in each S
j
remain balanced. The net change in each V
(i)
0
is the same for all i and each loses at least
h vertices because |T
′
| > |T |.
We repeat the process of creating W
(i)
j

, C
(i)
j
,
˜
G, and enlarging T = {S
1
, . . . , S
ℓ
} in
˜
G by
Lemma 4.7 until the number of vertices remaining in each exceptional set is less than 6h.
There is one caveat: If too many vertices are removed from the clusters of S
j
, then we
will not be able to apply the Blow-up Lemma later. Therefore, we introduce the following
notion: If in the entire process, at least d
1
L/3 (uncolored) vertices ar e removed from a
cluster C of S
j
, then both C and S
j
are called dead (otherwise live). The dead clusters
will be not considered until Step 5, after all the exceptional vertices have been removed.
The number o f dead cluster-triangles is not very large. To see this, there are three ways
for vertices to leave a cluster. First, they are placed in a K
h,h,h
with a vertex from the

exceptional set, so each vertex class V
(i)
loses at most

3
i=1
|V
(i)
0
|h vertices in this way.
Second, each time when we apply Lemma 4.7, there are at most 15 triangles in T
′
\ T
and there are a total of 15h vertices lost to 15 copies of K
h,h,h
. Third, there are at most 3
clusters not covered by T
′
and they could lo se 3h vertices to the exceptional sets. Since we
apply Lemma 4.7 at most |V
(i)
0
|/h times, the total number of vertices that leave clusters
is a t most
3|V
(i)
0
|h +

|V

(i)
0
|/h

(15h + 3h) = |V
(i)
0
|(3h + 18)  4ε
1
N(3h + 18).
the electronic journal of combinatorics 16 (2009), #R109 12
The number of dead cluster triangles is at most
4ε
1
N(3h + 18)
(d
1
/3)L

36(h + 6)ε
1
d
1
(1 −ε
1
)
ℓ <
d
1
2

ℓ.
because ε
1
≪ d
1
.
Because the number of dead clusters is not large, in the subgraph induced by live clusters,
each cluster is still reachable from every o t her cluster in the same partition class. Each
vertex in V
(i)
0
belongs to at least (1/3 − 3d
1
)ℓ live clusters. By letting U
′
be the set of
available live clusters, we still have |U
′
|  (1 − d
1
)ℓ when applying Claim 4.6. After
removing the edges incident with dead S
j
’s, the minimum-degree condition in
˜
G is still
¯
δ(
˜
G)  (2/3 −3d

1
)m and Lemma 4.7 can still be applied.
At the end each cluster (live or dead) has at least (1 −3ε
1
)L − d
1
L/3 uncolored vertices.
Each of the three clusters in any S
j
has t he same number of uncolored vertices, and this
number is always divisible by h.
Step 5: Insert the remaining exceptional vertices and apply t he Blow-up
Lemma
At this stage, the exceptional sets V
(i)
0
, i = 1, 2, 3 are all of the same size, divisible by h
and at most 5h (because it is less than 6h). Consider a vertex x ∈ V
(1)
0
and insert x into a
live cluster V
(1)
j
to which x belongs (as shown in Step 4, x belongs to at least (1/3 −3d
1
)ℓ
live clusters). As a result, V
(1)
j

loses h−1 vertices while V
(2)
j
and V
(3)
j
each loses h vertices.
To balance S
j
, we move a vertex from V
(1)
j
to V
(1)
1
following the remark in Step 3. As a
result, V
(1)
j
loses one orange vertex, and V
(1)
1
gains a n extra uncolored vertex. Repeat this
to all the vertices in V
(1)
0
∪ V
(2)
0
∪ V

(3)
0
. All S
j
, j > 1, have the same number of non-red
vertices among its three clusters. The same holds for S
1
because |V
(1)
0
| = |V
(2)
0
| = |V
(3)
0
|.
In addition, the number of non-red vertices in each cluster is at least (1 − d
1
/2)L, and
always a multiple of h.
Then, uncolor all the remaining orange vertices and r emove all red copies of K
h,h,h
from
G. Since each cluster now has at least (1 − d
1
/2)L vertices, by the Slicing Lemma, any
pair of clusters in S
j
is (ε

1
/2)-regular. Furthermore, each vertex in one cluster of S
j
is
adjacent to at least (d
1
−ε
1
)L −d
1
L/2 vertices in any other cluster of S
j
. Hence all pairs
in S
j
are (ε
1
/2, d
1
/3)-super-regular. We finally apply the Blow-up Lemma to each S
j
to
complete the K
h,h,h
-factor of G.
5 Conclud i ng Remarks
• We could reduce the error term γN in Theorem 1.2 to a constant C = C(h) by
showing that if G ∈ G
3
(N) is in the extreme case with sufficiently small γ and

the electronic journal of combinatorics 16 (2009), #R109 13
¯
δ(G)  2N/3 + C, then G contains a K
h,h,h
-factor. Unfortunately, the methods
involve a detailed case analysis which is too long to be included in this paper.
However, we can summarize them as follows. Given a positive integer h, let f (h)
be the smallest m for which there exists an N
0
such that every balanced tripartite
graph G ∈ G
3
(N) with N  N
0
, h divides N, and
¯
δ(G)  m contains a K
h,h,h
-factor.
Suppose that N = (6q + r)h with 0  r  5. Then, from Proposition 1.5 and a
manuscript [2 1] which details the proof of the extreme case:
f(h) =
2N
3
+ h − 1, if r = 0;
h

2N
3h


+ h − 2  f(h)  h

2N
3h

+ h − 1, if r = 1 , 2, 4, 5;
2N
3
+ h − 1  f(h) 
2N
3
+ 2h −1, if r = 3.
We have no conjecture as to whether the upper or lower bound is correct.
• The task of obtaining a tight minimum pairwise degree condition fo r K
r
-factors in
G
r
(N) becomes more challenging for larger r. The r = 2 case is very easy – we either
consider a maximum matching or apply the K¨onig-Hall theorem. The r = 3, 4 cases
become hard – [19] and [20] both applied the Regularity Lemma. At present a tight
Hajnal–Szemer´edi-type result is out of reach (though an approximate version was
given by Csaba and Mydlarz [5]).
• We believe one can prove a similar result as Theorem 1.2 for tiling 4 -colorable
graphs in 4-partite graphs by adopting the approach of [20] and the techniques in
this paper. In general, suppose that we know that every r-partite graph G ∈ G
r
(n)
with
¯

δ(G)  cn contains a K
r
-factor. Then applying the Regularity Lemma, one
can easily prove that for a ny ε > 0 and any r-colorable H, every G ∈ G
r
(n) with
¯
δ(G)  (c + ε)n contains an H-tiling that covers all but εn vertices (this is similar
to an early result of Alon and Yuster [1]). However, it is not clear how t o reduce
the number of leftover vertices to a constant, or zero (to get an H-factor). As seen
from the present manuscript, a minimum degree condition for K
r
-factors does not
immediately gives a similar degree condition for K
r
(h)-factors, where K
r
(h) is the
complete r-partite graph with h vertices in each partition set.
• Theorem 1.2 gives a near tight minimum degree condition
¯
δ  (2/3 + o(1))N for
K
h,h,h
-tilings. However, the coefficient 2/3 may not be best p ossible f or other 3-
colorable graphs, e.g., K
1,2,3
. In fact, when tiling a general (instead of 3-partite)
graph with certain 3-colorable H, the minimum degree threshold given by K¨uhn and
Osthus [1 7] has coefficient 1−1/χ

cr
(H) instead of 2/3, where χ
cr
(H) is the so-called
critical chromatic number. It would be interesting to see if something similar holds
for tr ipart ite tiling.
the electronic journal of combinatorics 16 (2009), #R109 14
Acknowledgments
The authors would like to thank the Department of Mathematics, Statistics, and Com-
puter Science at the University of Illinois at Chicago for their supporting the first author
via a visitor fund. The author also thanks a referee for her/his suggestions that improved
the presentation.
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